The system under study in this work is a self-igniting pile of solid material. To predict and understand the effect of steep changes of the state variables on such systems, a reaction-diffusion model is employed. These systems can exhibit complex oscillatory behaviour, and changes in ambient conditions over time may strongly impact the inherent oscillations. To simulate the unsteady evolution of the pile, both a classical numerical technique (method of lines) and a reduced order approach are employed in combination with a stiff ODE solver. To account for circadian fluctuations in temperature, time-variable boundary conditions are assumed upon formulating the problem. The reduced order model is introduced in view of understanding if an approximated formulation characterized by a much lower number of state variables can accurately predict the complex behaviour of the system even in the case of sudden, steep variations of the values of the state variables due to the phenomenon of self-ignition, intensified here by variable boundary conditions. The selected case studies have the goal of exploring the effect of stockpile properties on the self-ignition phenomenon. Numerical solutions show the anticipated coupling between the system intrinsic dynamics and the oscillating temperature imposed at the boundary. All of the analysed cases are accurately replicated by the reduced order model.